How much does light bend?

We have an explanation for why light bends when it moves across a boundary between two different media but is there any predictable pattern to the amount that any given ray of light bends?

I’m going to start this explanation with an apology to purists who might complain that the following explantion of how light bends is either incomplete or untrue. Yes, I know that but it’s only meant to be a simple introduction to the concepts! And I’ve provided a reference at the end for those who want more meat on the bone.

As the diagram above shows, rays of light striking a curved lens surface at different distances from the central axis encounter the boundary at an increasingly steep angle. This in turn means the rays of light bend by increasingly large amounts. The incident angle i2 is greater than the incident angle i1 so the refracted angle r2 will also be greater than the refracted angle r1. If we are lucky, the relationship between the angles r1 and r2 will be such that the amount by which the rays of light bend will vary such that the rays will meet at a point on the optical axis – and every other ray of light that arrives at the front surface of the lens will also be directed to this same spot. The distance to that spot is known as the focal length of the lens.

Of course, lenses have two surfaces and if we assume they are both curved then the overall effect looks something like this…

In this diagram the two surfaces have slightly different radii of curvature, which is often the case in practice.

It happens that the “lucky” situation mentioned above, where the rays of light all meet at one point on the optical axis, owes nothing at all to luck. Back in the Sixteenth Century, pre-photography but amid a massive interest in astronomy, The Lens Makers Formula was born; this links the focal length (f) with the refractive index of the glass (n) and the radii of curvature of the two surfaces (R1 and R2) as follows…

1/f = (n – 1)(1/R1 – 1/R2)

We don’t need to worry about the mathematics here but if any reader wants to find out more then I highly recommend the book Fundamentals of Optics by Francis Jenkins and Harvey White (McGraw Hill).

In the next instalment we’ll look at how the focal length of the lens allows an image to be formed.